Question

what must be marked on the triangles below for them to be congruent by sss?
what must be marked on the triangles below for them to be congruent by asa?

Explanation:

Step1: Analyze the first figure

In the first figure, we have two sides equal (marked with one tick each) and one non - included angle equal. So, it is not SSS (Side - Side - Side), not SAS (Side - Angle - Side) as the angle is non - included, AAS (Angle - Angle - Side) is not applicable here, ASA (Angle - Side - Angle) is not applicable as the side is not between the angles, and HL (Hypotenuse - Leg, for right - triangles) is not applicable. So, the answer for the first 'Yes/No' is 'No' for all SSS, SAS, AAS, ASA, HL.

Step2: Analyze the second figure

In the second figure, we have two sides equal (marked with one tick each). But we don't have enough information about the angles or the third side to apply SSS, SAS (the angle between the sides is not given as equal), AAS (not enough angles marked), ASA (not enough angles and side in between marked), and HL (not right - triangles). So, the answer for the second 'Yes/No' is 'No' for all SSS, SAS, AAS, ASA, HL.

Step3: Analyze the third figure

In the third figure, we have two sides equal (31 and 42). The third sides are 40. By SSS (if all three sides are equal: \(QR = KG = 31\), \(QS=HG = 42\), \(RS = KH=40\)), the triangles are congruent. So, for the third 'Yes/No', the answer is 'Yes' for SSS and 'No' for SAS, AAS, ASA, HL.

Step4: Determine what to mark for SSS in the fourth figure

For two triangles \(\triangle RQS\) and \(\triangle DQS\) to be congruent by SSS, since \(RQ = DQ\) (marked) and \(QS\) is common. We need \(RS = DS\) to be marked.

Step5: Determine what to mark for ASA in the fifth figure

For two triangles \(\triangle DEF\) and \(\triangle LKJ\) to be congruent by ASA (Angle - Side - Angle), since \(\angle D=\angle L\) (assume we know one pair of angles) and \(\angle E=\angle K\) (marked). We need \(EF = KJ\) (the side between the two angles) to be marked.

Answer:

• First figure: No for SSS, No for SAS, No for AAS, No for ASA, No for HL.
• Second figure: No for SSS, No for SAS, No for AAS, No for ASA, No for HL.
• Third figure: Yes for SSS, No for SAS, No for AAS, No for ASA, No for HL.
• Fourth figure: \(RS = DS\) must be marked.
• Fifth figure: \(EF = KJ\) must be marked.